A generalization of a theorem of Beardon on analytic contraction mappings
type: Item
instance of: scholarly article
title: A generalization of a theorem of Beardon on analytic contraction mappings
zbMATH Open document ID: 0870.30022
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-09-07T00:00:00Z
review text: Let \(D\) be a bounded convex domain in \(\mathbb{C}\), and let \(\mathcal F\) be a contracting family of analytic self-mappings of \(D\), i.e., \(|f'(z)|\leq 1\) for \(z\in D\) and \(f\in{\mathcal F}\), and \(\sup_{f\in{\mathcal F}}|f'(z_0)|= 1-c_0\) for some \(z_0\in D\) and some \(c_0>0\). Further, let \(F_n= f_1\circ f_2\circ\cdots\circ f_n\) for \(n\in\mathbb{N}\) and \(f_1,\dots,f_n\in{\mathcal F}\). The author considers the question whether each sequence \((F_n)_{n\in\mathbb{N}}\) converges uniformly to a constant. The key to get such convergence results is to show that the Euclidean diameter of \(F_n(D)\) tends to \(0\). \textit{A. F. Beardon} [Proc. Int. Conf. on Computational Methods and Function Theory, eds. R. M. Ali, St. Rusheweyh and E. B. Saff, 111-128 (1995; per bibl.)] has given the explicit estimate \(\text{diam }F_n(D)= O(1/n^{1/2})\) under the assumption that the curvature of \(\partial D\) is uniformly bounded away from both \(0\) and \(\infty\). For in some sense less strong conditions on \(\partial D\), the author derives an estimate for \(\text{diam }F_n(D)\), which gives, in particular, \(\text{diam }F_n(D)\to 0\), and includes Beardon's result.
reviewed by: N.Terglane
Mathematics Subject Classification ID: 30D05
zbMATH DE Number: 902532
zbMATH Keywords: composition of analytic mappings
author: Philip Jonathan Rippon
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0133
OpenAlex ID: W1976114216
DOI: 10.1006/JMAA.1996.0133
description: scientific article; zbMATH DE number 902532